Searching for an asymptotic to exp[0.5]

[sheldonison]

Im searching for an asymptotic to exp[0.5](x).

This problem will probably keep me occupied for a long time. I need to work on tools for figuring out the growth from the Taylor series, where growth is the limit of slog(f^n(x0))/n (I have some ideas).

I'm also thinking about what the Taylor series for an entire pseudo half iterate might look like, and what bounds can be put on the Taylor series coefficients. The pseudo half iterate of exp should converge with far fewer Taylor series terms than exp(z). half(10000)~=1E22, which is only 10000^5.5 Anyway, I expect this problem will keep me occupied for a long time, but that also means it might take awhile to make any real progress...

edit: Emperical testing suggests that an "entire" pseudo half iterate is very likely possible, with all positive Taylor series coefficients at z=0, and a probable growth value of 0.5, as defined by the "growth" equation. I can post the empirical results for the first 100 derivatives of such a conjectured asymptotic solution later. Each derivative is bounded to a maximum value by a particular value of half(z). I can post more later; still working on how to formalize the definition of the conjectured Taylor series.
- Sheldon